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CAIE M2 2005 November Q7
11 marks Standard +0.3
7 A particle of mass 0.25 kg moves in a straight line on a smooth horizontal surface. A variable resisting force acts on the particle. At time \(t \mathrm {~s}\) the displacement of the particle from a point on the line is \(x \mathrm {~m}\), and its velocity is \(( 8 - 2 x ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It is given that \(x = 0\) when \(t = 0\).
  1. Find the acceleration of the particle in terms of \(x\), and hence find the magnitude of the resisting force when \(x = 1\).
  2. Find an expression for \(x\) in terms of \(t\).
  3. Show that the particle is always less than 4 m from its initial position. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
    University of Cambridge International Examinations is part of the University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE M2 2006 November Q1
4 marks Moderate -0.8
1 A stone is projected horizontally with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The horizontal and vertically upward displacements of the stone from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Find the equation of the stone's trajectory. The stone enters the sea at a horizontal distance of 24 m from the base of the cliff.
  2. Find the height above sea level of the top of the cliff.
CAIE M2 2006 November Q2
4 marks Moderate -0.3
2 A horizontal turntable rotates with constant angular speed \(3 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A particle of mass 0.06 kg is placed on the turntable at a point 0.25 m from its centre. The coefficient of friction between the particle and the turntable is \(\mu\). As the turntable rotates, the particle moves with the turntable and no sliding takes place.
  1. Find the vertical and horizontal components of the contact force exerted on the particle by the turntable.
  2. Show that \(\mu \geqslant 0.225\).
CAIE M2 2006 November Q3
5 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-2_892_412_1217_865} A hollow cylinder of radius 0.35 m has a smooth inner surface. The cylinder is fixed with its axis vertical. One end of a light inextensible string of length 1.25 m is attached to a fixed point \(O\) on the axis of the cylinder. A particle \(P\) of mass 0.24 kg is attached to the other end of the string. \(P\) moves with constant speed in a horizontal circle, in contact with the inner surface of the cylinder, and with the string taut (see diagram).
  1. Find the tension in the string.
  2. Given that the magnitude of the acceleration of \(P\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the force exerted on \(P\) by the cylinder.
CAIE M2 2006 November Q4
5 marks Standard +0.3
4 A stone is projected from a point on horizontal ground with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal, where \(\sin \theta = \frac { 4 } { 5 }\). At time 1.2 s after projection the stone passes through the point \(A\). Subsequently the stone passes through the point \(B\), which is at the same height above the ground as \(A\). Find the horizontal distance \(A B\).
CAIE M2 2006 November Q5
6 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_383_1031_543_557} A non-uniform rod \(A B\) of length 2.5 m and mass 3 kg has its centre of mass at the point \(G\) of the rod, where \(A G = 1.5 \mathrm {~m}\). The rod hangs horizontally, in equilibrium, from strings attached at \(A\) and \(B\). The strings at \(A\) and \(B\) make angles with the vertical of \(\alpha ^ { \circ }\) and \(15 ^ { \circ }\) respectively. The tension in the string at \(B\) is \(T \mathrm {~N}\) (see diagram). Find
  1. the value of \(T\),
  2. the value of \(\alpha\).
CAIE M2 2006 November Q6
8 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-3_597_690_1416_731} A large uniform lamina is in the shape of a right-angled triangle \(A B C\), with hypotenuse \(A C\), joined to a semicircle \(A D C\) with diameter \(A C\). The sides \(A B\) and \(B C\) have lengths 3 m and 4 m respectively, as shown in the diagram.
  1. Show that the distance from \(A B\) of the centre of mass of the semicircular part \(A D C\) of the lamina is \(\left( 2 + \frac { 2 } { \pi } \right) \mathrm { m }\).
  2. Show that the distance from \(A B\) of the centre of mass of the complete lamina is 2.14 m , correct to 3 significant figures.
CAIE M2 2006 November Q7
9 marks Challenging +1.2
7 A cyclist starts from rest at a point \(O\) and travels along a straight path. At time \(t \mathrm {~s}\) after starting, the displacement of the cyclist from \(O\) is \(x \mathrm {~m}\), and the acceleration of the cyclist is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(a = 0.6 x ^ { 0.2 }\).
  1. Find an expression for the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of the cyclist in terms of \(x\).
  2. Show that \(t = 2.5 x ^ { 0.4 }\).
  3. Find the distance travelled by the cyclist in the first 10 s of the journey.
CAIE M2 2006 November Q8
9 marks Challenging +1.2
8 \includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-4_933_275_689_934} The diagram shows a light elastic string of natural length 0.6 m and modulus of elasticity 5 N with one end attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at the point \(A\), which is 0.9 m vertically above \(O\). The particle is released from rest and travels vertically downwards through \(O\) to the point \(C\), where it starts to move upwards. \(B\) is the point of the line \(A C\) where the string first becomes slack.
  1. Find the speed of \(P\) at \(B\).
  2. The extension of the string when \(P\) is at \(C\) is \(x \mathrm {~m}\).
    (a) Show that \(x ^ { 2 } - 0.48 x - 0.81 = 0\).
    (b) Hence find the distance \(A C\).
CAIE M2 2007 November Q1
5 marks Standard +0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_711_398_269_877} Each of two identical light elastic strings has natural length 0.25 m and modulus of elasticity 4 N . A particle \(P\) of mass 0.6 kg is attached to one end of each of the strings. The other ends of the strings are attached to fixed points \(A\) and \(B\) which are 0.8 m apart on a smooth horizontal table. The particle is held at rest on the table, at a point 0.3 m from \(A B\) for which \(A P = B P\) (see diagram).
  1. Find the tension in the strings.
  2. The particle is released. Find its initial acceleration.
CAIE M2 2007 November Q2
6 marks Moderate -0.3
2 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-2_496_609_1535_769} One end of a light inextensible string of length 0.16 m is attached to a fixed point \(A\) which is above a smooth horizontal table. A particle \(P\) of mass 0.4 kg is attached to the other end of the string. \(P\) moves on the table in a horizontal circle, with the string taut and making an angle of \(30 ^ { \circ }\) with the downward vertical through \(A\) (see diagram). \(P\) moves with constant speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  1. the tension in the string,
  2. the force exerted by the table on \(P\).
CAIE M2 2007 November Q3
7 marks Standard +0.3
3 \includegraphics[max width=\textwidth, alt={}, center]{b9080e9f-2c23-43ce-b171-bd68648dc56b-3_764_627_274_758} A uniform beam \(A B\) has length 2 m and mass 10 kg . The beam is hinged at \(A\) to a fixed point on a vertical wall, and is held in a fixed position by a light inextensible string of length 2.4 m . One end of the string is attached to the beam at a point 0.7 m from \(A\). The other end of the string is attached to the wall at a point vertically above the hinge. The string is at right angles to \(A B\). The beam carries a load of weight 300 N at \(B\) (see diagram).
  1. Find the tension in the string. The components of the force exerted by the hinge on the beam are \(X \mathrm {~N}\) horizontally away from the wall and \(Y \mathrm {~N}\) vertically downwards.
  2. Find the values of \(X\) and \(Y\).
CAIE M2 2007 November Q4
7 marks Standard +0.3
4 A particle of mass 0.4 kg is released from rest and falls vertically. A resisting force of magnitude \(0.08 v \mathrm {~N}\) acts upwards on the particle during its descent, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the particle at time \(t \mathrm {~s}\) after its release.
  1. Show that the acceleration of the particle is \(( 10 - 0.2 v ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find the velocity of the particle when \(t = 15\).
CAIE M2 2007 November Q5
7 marks Challenging +1.2
5 Each of two light elastic strings, \(S _ { 1 }\) and \(S _ { 2 }\), has modulus of elasticity 16 N . The string \(S _ { 1 }\) has natural length 0.4 m and the string \(S _ { 2 }\) has natural length 0.5 m . One end of \(S _ { 1 }\) is attached to a fixed point \(A\) of a smooth horizontal table and the other end is attached to a particle \(P\) of mass 0.5 kg . One end of \(S _ { 2 }\) is attached to a fixed point \(B\) of the table and the other end is attached to \(P\). The distance \(A B\) is 1.5 m . The particle \(P\) is held at \(A\) and then released from rest.
  1. Find the speed of \(P\) at the instant that \(S _ { 2 }\) becomes slack.
  2. Find the greatest distance of \(P\) from \(A\) in the subsequent motion.
CAIE M2 2007 November Q6
9 marks Standard +0.2
6 A particle is projected from a point \(O\) at an angle of \(35 ^ { \circ }\) above the horizontal. At time \(T\) s later the particle passes through a point \(A\) whose horizontal and vertically upward displacements from \(O\) are 8 m and 3 m respectively.
  1. By using the equation of the particle's trajectory, or otherwise, find (in either order) the speed of projection of the particle from \(O\) and the value of \(T\).
  2. Find the angle between the direction of motion of the particle at \(A\) and the horizontal. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_476_895_269_625} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Fig. 1 shows the cross-section of a uniform solid. The cross-section has the shape and dimensions shown. The centre of mass \(C\) of the solid lies in the plane of this cross-section. The distance of \(C\) from \(D E\) is \(y \mathrm {~cm}\).
  3. Find the value of \(y\). The solid is placed on a rough plane. The coefficient of friction between the solid and the plane is \(\mu\). The plane is tilted so that \(E F\) lies along a line of greatest slope.
  4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_375_431_1366_897} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The solid is placed so that \(F\) is higher up the plane than \(E\) (see Fig. 2). When the angle of inclination is sufficiently great the solid starts to topple (without sliding). Show that \(\mu > \frac { 1 } { 2 }\). [3]
  5. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b9080e9f-2c23-43ce-b171-bd68648dc56b-5_376_428_2069_900} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} The solid is now placed so that \(E\) is higher up the plane than \(F\) (see Fig. 3). When the angle of inclination is sufficiently great the solid starts to slide (without toppling). Show that \(\mu < \frac { 5 } { 6 }\). [3]
CAIE M2 2008 November Q1
4 marks Standard +0.3
1 One end of a light elastic rope of natural length 2.5 m and modulus of elasticity 80 N is attached to a fixed point \(A\). A stone \(S\) of mass 8 kg is attached to the other end of the rope. \(S\) is held at a point 6 m vertically below \(A\) and then released. Find the initial acceleration of \(S\).
CAIE M2 2008 November Q2
4 marks Standard +0.8
2 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-2_485_863_495_641} A uniform solid cylinder has height 24 cm and radius \(r \mathrm {~cm}\). A uniform solid cone has base radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder and the cone are both placed with their axes vertical on a rough horizontal plane (see diagram, which shows cross-sections of the solids). The plane is slowly tilted and both solids remain in equilibrium until the angle of inclination of the plane reaches \(\alpha ^ { \circ }\), when both solids topple simultaneously.
  1. Find the value of \(h\).
  2. Given that \(r = 10\), find the value of \(\alpha\).
CAIE M2 2008 November Q3
7 marks Moderate -0.3
3 A particle \(P\) of mass 0.5 kg moves along the \(x\)-axis on a horizontal surface. When the displacement of \(P\) from the origin \(O\) is \(x \mathrm {~m}\) the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Two horizontal forces act on \(P\); one force has magnitude \(\left( 1 + 0.3 x ^ { 2 } \right) \mathrm { N }\) and acts in the positive \(x\)-direction, and the other force has magnitude \(8 \mathrm { e } ^ { - x } \mathrm {~N}\) and acts in the negative \(x\)-direction.
  1. Show that \(v \frac { \mathrm {~d} v } { \mathrm {~d} x } = 2 + 0.6 x ^ { 2 } - 16 \mathrm { e } ^ { - x }\).
  2. The velocity of \(P\) as it passes through \(O\) is \(6 \mathrm {~ms} ^ { - 1 }\). Find the velocity of \(P\) when \(x = 3\).
  3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-3_259_745_278_740} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A small sphere \(A\) of mass 0.15 kg is moving inside a fixed smooth hollow cylinder whose axis is vertical. \(A\) moves with constant speed \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a horizontal circle of radius 0.35 m , and is continuously in contact with both the plane base and the curved surface of the cylinder. Fig. 1 shows a vertical cross-section of the cylinder through its axis. Find the magnitude of the force exerted on \(A\) by
CAIE M2 2008 November Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{5109244c-3062-4f5f-9277-fc6b5b28f2d4-4_495_1405_264_370} \(A B C D\) is a central cross-section of a uniform rectangular block of mass 35 kg . The lengths of \(A B\) and \(B C\) are 1.2 m and 0.8 m respectively. The block is held in equilibrium by a rope, one end of which is attached to the point \(E\) of a rough horizontal floor. The other end of the rope is attached to the block at \(A\). The rope is in the same vertical plane as \(A B C D\), and \(E A B\) is a straight line making an angle of \(20 ^ { \circ }\) with the horizontal (see diagram).
  1. Show that the tension in the rope is 187 N , correct to the nearest whole number.
  2. The block is on the point of slipping. Find the coefficient of friction between the block and the floor.
CAIE M2 2008 November Q6
9 marks Standard +0.3
6 A light elastic string has natural length 4 m and modulus of elasticity 2 N . One end of the string is attached to a fixed point \(O\) of a smooth plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass \(0.1 \mathrm {~kg} . P\) is held at rest at \(O\) and then released. The speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the extension of the string is \(x \mathrm {~m}\).
  1. Show that \(v ^ { 2 } = 45 - 5 ( x - 1 ) ^ { 2 }\). Hence find
  2. the distance of \(P\) from \(O\) when \(P\) is at its lowest point,
  3. the maximum speed of \(P\).
CAIE M2 2008 November Q7
10 marks Standard +0.3
7 A particle \(P\) is projected from a point \(O\) on horizontal ground with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and direction \(60 ^ { \circ }\) upwards from the horizontal. At time \(t \mathrm {~s}\) later the horizontal and vertical displacements of \(P\) from \(O\) are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Write down expressions for \(x\) and \(y\) in terms of \(V\) and \(t\) and hence show that the equation of the trajectory of \(P\) is $$y = ( \sqrt { } 3 ) x - \frac { 20 x ^ { 2 } } { V ^ { 2 } }$$ \(P\) passes through the point \(A\) at which \(x = 70\) and \(y = 10\). Find
  2. the value of \(V\),
  3. the direction of motion of \(P\) at the instant it passes through \(A\).
CAIE M2 2009 November Q1
3 marks Moderate -0.5
1 \includegraphics[max width=\textwidth, alt={}, center]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_113_787_264_680} A light elastic spring of natural length 0.25 m and modulus of elasticity 100 N is held horizontally between two parallel plates. The axis of the spring is at right angles to each of the plates. The horizontal force exerted on the spring by each of the plates is 20 N (see diagram). Find the amount by which the spring is compressed and hence write down the distance between the plates.
CAIE M2 2009 November Q2
5 marks Standard +0.3
2 A particle of mass 0.2 kg is attached to one end of a light elastic string of natural length 0.6 m and modulus of elasticity 4 N . The other end of the string is attached to a fixed point \(O\). The particle is held at a point which is \(( 0.6 + x ) \mathrm { m }\) vertically below \(O\). The particle is released from rest. In the subsequent motion the speed of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when the string becomes slack. By considering energy, find the value of \(x\).
CAIE M2 2009 November Q3
6 marks Standard +0.3
3 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_408_291_1027_927} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A uniform solid cylinder has mass 8 kg and height 16 cm . A uniform solid cone, whose base radius is the same as the radius of the cylinder, has mass 2 kg and height 12 cm . A composite solid is formed by joining the cylinder and cone so that the base of the cone coincides with one end of the cylinder (see Fig. 1).
  1. Show that the centre of mass of the composite solid is 10.2 cm from its base. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{68acf474-5da2-4949-b3b2-fc42cd73bd4a-2_401_444_1877_849} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure} The composite solid is held with a point on the circumference of its base in contact with a horizontal table. The base makes an angle \(\theta ^ { \circ }\) with the table (see Fig. 2, which shows a cross-section). When the cone is released it moves towards the equilibrium position in which its base is in contact with the table.
  2. Given that the radius of the base is 4 cm , find the greatest possible value of \(\theta\), correct to 1 decimal place.
CAIE M2 2009 November Q4
7 marks Standard +0.3
4 A particle is projected from a point \(O\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta\) above the horizontal. After 0.3 s the particle is moving with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\tan ^ { - 1 } \left( \frac { 7 } { 24 } \right)\) above the horizontal.
  1. Show that \(V \cos \theta = 24\).
  2. Find the value of \(V \sin \theta\), and hence find \(V\) and \(\theta\).